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2 Contents Bernoulli Riccati λ λx, y Clairaut Lagrange d Alembert n Bessel Legendre Type Type Type Type Euler i

3 ii CONTENTS Vector Introduction Fourier 66 1 Fourier Fourier data Fourier data Fourier line of swiftest descent 79 1 I Method of Straightforward Expansion Method of Strained Coordinates Parameters Method of Multiple Scales II ii -

4 CONTENTS iii Duffing Satellite Rayleigh Van del Pol III Laplace Equation Introduction Laplace Laplace Laplace Laplace Laplace Laplace Legendre Legendre Bessel Runge-Kutta Runge-Kutta Runge-Kutta Runge-Kutta-Gill Euclid / Cardano Ferrari D iii -

5 iv CONTENTS 1.1 Square Matrix matrix Square Matrix Square Matrix N m, n Eigen vector Eigenvalue for 3 3 matrix Eigenvalue for 4 4 matrix iv -

6 Chapter / in TEX/ x, y y yx f x, y, dy = 0 dx dy = F x, y dx normal form Fourier Fourier Laplace 1.1 dy dy = P xqy dx Qy = P x dx + C C = constant 1. dy y dx = f x du fu u = dx x 1.3 dy y = ux dx = u + xdu dx [ ] du fu u x = C exp 1 1 dy dy + P xy = Qx dx dx + P xy = 0 [ y = C exp ] P x dx [ y = exp ] { P x dx [ Qx exp ] } P x dx dx 1

7 CHAPTER 1. = + Lagrange 1.4 Bernoulli dy dx + P xy + Qxya = 0 a 1 u = y 1 a 1 du + P xu = Qx 1 a dx 1.5 Riccati dy dx + P xy + Qxy = Rx 1 1 Qx = 0 1 Qx 0 y = 1 dv Qxv dx v d [ v dx + P x 1 ] dq dv Q dx dx QxRxv = x, y P x, y dx + Qx, y dy = 0 Q x = P y P x, y dx + Qx, y dy = dφ = 0 : φ = constant Laplace 1.7 λ λx, y x, y P x, y dx + Qx, y dy = 0 λp x, y dx + λqx, y dy = 0

8 x λq = y λp λp x, y dx + λqx, y dy = dφ = 0 : φ = constant λ 1 Laplace Clairaut x, y yx y = x dy dy dx + f dx TYPE a: P = dy dx dy dx = P + xdp dx + dp df dx dp y = xp + fp x + df dp dp dx = 0 zero TYPE a, b dp = 0, dx y = xp + fp P = constant C, y = Cx + fc TYPE b: x + df = 0, dp y = xp + fp TYPE b x P x = df/dp y P TYPE a C TYPE b TYPE b singular solution TYPE b TYPE a y = Cx + fc envelope 1.9 Lagrange d Alembert dy dy y = xφ + ψ P = dy dx dx dx y = xφp + ψp P = φp + x dp dφ dx dp + dp dψ dx dp [φp P ] dx dp + dφ dp x = dψ dp φp P P, x 1 x = CfP + gp fp gp y = xφp + ψp Lagrange φp = P Clairaut 3

9 4 CHAPTER n n n 1 dy dy + Q 1 x, y + + Q n 1x, y dy dx dx dx + Q nx, yy = 0 P = dy/dx P n n R 1 x, y, R x, y,, R n 1 x, y, R n x, y [P R 1 x, y] [P R x, y] [P R n 1 x, y] [P R n x, y] = 0 1 n n 1 1 P = R 1 x, y, P = R x, y,, P = R n 1 x, y, P = R n x, y n n Φ 1 x, y, C = 0, Φ x, y, C = 0,, Φ n 1 x, y, C = 0, Φ n x, y, C = 0 C: 1 n Φ 1 x, y, C Φ x, y, C Φ n 1 x, y, C Φ n x, y, C = 0 [ ] 1 P xp y = 0 P dy dx y x C [y C expx] = fx, y, P = 0 P dy/dx 3 x, y, P y-p P = gx, y 4

10 . 5 x, y yx f x, y, dy dx, d y d y = 0 dx dx = g x, y, dy dx 1 d y dx + P xdy + Qxy = Rx dx.1 d y dx + ady dx + by = 0 y expλx λ λ + aλ + b = 0 λ 1, λ y = C 1 expλ 1 x + C expλ x C 1, C : λ = a/ b = a /4 y = C 1 + C x expλx y = C 1 y 1 + C y y 1 = expλ 1 x, y = expλ x y 1 = expλx, y = x expλx A 1, A y 1, y A 1 y 1 + A y = 0 dy 1 A 1 dx + A dy dx = 0 A 1, A A 1, A Wronskian D Dx = Dy 1, y ; dy 1 /dx, dy /dx D = y 1 y dy 1 dx zero D 0 A 1, A A 1 = A = 0 dy dx y 1, y D D 0 y 1, y 5

11 6 CHAPTER 1. y 1, y y 1, y D = 0 D 0 [ ] Wronskian D Dx = Dy 1, y ; dy 1 /dx, dy /dx dd dx = D dy1 dx, dy dx ; dy 1 dx, dy + D y 1, y ; dx d y 1 = D y 1, y ; dx, d y dx d y 1 dx, d y dx d y 1 D y 1, y ; dx, d y = D y dx 1, y ; a dy 1 dx by 1, a dy dx by dy 1 = ad y 1, y ; dx, dy dx D 1 dd dx = ad x = 0 Wronskian D0 Dx = D0 exp ax ax D0 0 x 0 Dx 0 D0 = 0 x 0 Dx = 0 Dx zero : y = C 1 y 1 + C y x = 0 y = y 0, dy/dx = v 0 y 0, v 0 nonzero y 0 = C 1 y C y 0, v 0 = C 1 dy1 dx x=0 + C dy dx C 1, C D0 0 Dx 0 x=0. d y dx + ady + by = Rx dx 6

12 .3 Bessel d y dx + 1 dy x dx + 1 n y = 0 n 0 x n Bessel n Bessel III VI Bessel.4 Legendre [ d 1 ] x dy + nn + 1y = 0 n 0 dx dx n Legendre 1 III V.5 III.6 d y dx + qxdy + Qxy = Rx dx P = dy dx, dp dx + qxp + Qxy = Rx y, P 1 3 N 1 dy 1 dx = f 1x, y 1, y,, y N 1, y N, dy dx = f x, y 1, y,, y N 1, y N,,,,,,, 7

13 8 CHAPTER 1. dy N 1 dy N = f N 1 x, y 1, y,, y N 1, y N, dx dx = f Nx, y 1, y,, y N 1, y N N 1 equilibrium solution y n = Y n = constant for n = 1,,, N z n z n x N 1 dz n dx = N m=1 P nm z m for n = 1,,, N, where P nm P nm x = f n y m P nm Y n N m m z 1 x, z m x,, z N 1x, m z m N x D Dx: z 1 1 x z 1 x z N 1x 1 z 1 N x z 1 x z x z N 1x z N x Dx = z N 1 1 x z N 1 x z N 1 N 1 x z N 1 N x z N 1 x z N x z N 1x N z N N x nonzero N Dx 0 N 1 D x D dd dx = P xd, where P x N P nn n=1 [ x ] Dx = D0 exp P x dx 0 P x x = 0 x 0 D0 0 Dx 0 N 3.1 P nm A nm = P nm N 1 A nm z n expλx for n = 1,,, N λ A 1 1 λ A 1 A 1 N 1 A 1 N A 1 A λ A N 1 A N Dλ = = 0 A N 1 1 A N 1 A N 1 N 1 λ A N 1 N A N 1 A N A N N 1 A N N λ 8

14 4. 9 λ N λ 1, λ,, λ N N z n = C nm expλ m x m=1 for n = 1,,, N C nm λ m expλ m x C nm N 4 n f x, y, dy dx, d y dx,, dn y = 0 dx n d n y dx = F x, y, dy n dx, d y dx,, dn 1 y = 0 n dx n Type 1 d n y dx n = fx n n 4. Type d n y dx n = fy n = 1 n = 1 1 dy = dx fy dy + C fy 1 :fy = a 0 + a 1 y a 0, a 1 : fy 3 fy y I III n > 9

15 10 CHAPTER Type 3 f x, dy dx, d y dx,, dn y dx n = 0 F x, P, dp dx, d P dx,, dn 1 P dx n 1 P = dy/dx P n 1 = Type 4 f y, dy dx, d y dx,, dn y dx n = 0 F y, P, dp dy, d P dy,, dn 1 P dy n 1 P = dy/dx y P n 1 = Euler N n=0 a n x n dn y dx n = Qx : x = expt t, yt N n=0 b n d n y dt n = Qexpt d y dx + y = 0 y = C 1 sinx + C cosx x = x 0 y dy/dx x = a x = b y a x b 10

16 i ya = 0, yb = 0 dy dy ii ya = yb, = dx dx x=a x=b dy dy iii αya = 0, βyb = 0 Strum dx dx x=a x=b a x b d y dx + pxdy dx + qxy = 0 [ ] [ P x = exp pxdx, Qx = qx exp [ d P x dy ] dx dx + Qxy = 0 self-adjoint differential equation ] pxdx 5. x, y y yx Ly Ly px d y dx + qxdy + rxy 5.1 dx z zx [ ] zly dx = z px d y dx + qxdy dx + rxy dx = z px dy dx dy d dx dx [z px] dx + z qx y y d dx [z qx] dx + z rx y dx = y Mz dx + Ny, z 5. Mz Ny, z Mz d d [z px] [z qx] + z rx, dx dx 5.3a Ny, z z px dy dx y d [z px] + z qx y dx 5.3b 11

17 Chapter D ,7.10 / addition / in TEX / review Linearity Linear Operator Operator L x, y t x xt, y yt Lax = alx a :, Lx + y = Lx + Ly L L x : x = X + ft X Xt ft: L X L non-linear [ ] L Lx x Lx = 0 Lx x x : vector Lx = 0 1 Lx dx Lx = 0 dt Lx x dt Lx = 0 d/dt L Lx = d x/dt + x Lx = 0 Linear Homogeneous Ordinary Differential Equation Lx = qt qt : inhomogeneous 1

18 quadrature Fourier Laplace Fourier 13 13

19 14 CHAPTER. 1 Linear Homogeneous First Order Ordinary Differential Equation x xt t pt: dx dt + ptx = 0 dx x = pt dt Lnx = pt dt + constant x = C exp[ pt dt] 1.1 C exp[ pt dt] fundamental solution 1.1 general solution 1 1 pt = constant = a a: 1.1 x = C exp at 1. x expλt λ: dx dt + ax = λ + ax = 0 λ + a = 0 x = 0 x 0 λ characteristic equation eigenvalue equation λ = a, assuming x 0 x x = C expλt = C exp at C : Linear Inhomogeneous First Order Ordinary Differential Equation x xt t pt, qt: dx dt + ptx = qt

20 1.4 x 0 : x 0 exp[ pt dt] 1.4 x = Xx 0 X Xt: 15 dx dt + ptx = dx dt x 0 + dx 0 dt X + ptxx 0 = dx dt x 0 = qt X x [ qt ] qt X = dt + constant x = x 0 dt + constant x 0 x [ 1 : x 0 x 1 0 qt dt ] particular solution : constant x variation of constants t = 0 x 1.5 x = x 0 [ t 0 qt x 0 dt + constant ] 1.6 Linear Homogeneous Second Order Ordinary Differential Equation with Constant Coefficients x xt t a: a 0 dx/dt t [ dx d ] x dt dt + axdx dt = dt d x + ax = dt dx dx dt d + dt ax dx = constant E 1 dx + ax = E dt dx = ± dt E ax E ax 0 x = E a sin a t + constant 1.8 a < 0 x = E/a sinh a t + constant sin a t cos a t 15

21 16 CHAPTER. 1.7 E constant 1.7 x expλt λ : λ x d x dt + ax = λ + a x = 0 λ = ± a, assuming x 0 x = C expi a t + C exp i a t 1.9a x = A sin a t + B cos a t 1.9b C, C : C C i 1 A, B: a < 0 expi a t, exp i a t, sin a t, cos a t 1.9a 1.9b expi a t exp i a t sin a t cos a t 1 x 0 1 x 1 x 0 = sin a t, x 1 = cos a t x 0 x 1 d x 0 dt + ax 0 = 0, d x 1 dt + ax 1 = a, b 1.10a x b x 0 d x 0 d x 1 dt + ax x 1 0 x 0 dt + ax 1 = d dx 0 x 1 dt dt x dx 1 0 = 0 dt dx 0 x 1 dt x dx 1 0 = constant A A dt x 0 x x 1 = Ax 0 x 0 dt 1.1 A 1.1 x 1 1 [ ] x 0 = sin a t x 1 = cos a t C 0, C 1 C 0 x 0 + C 1 x 1 = 0 16

22 C 0 = C 1 = 0 x 0 x 1 x 0 x 1 dx 0 C 0 dt + C dx 1 1 dt = 0 C 0, C 1 C 0, C 1 W : dx 1 W = x 0 dt x dx 0 1 dt zero W 0 C 0 = C 1 = 0 W W = a sin a t a cos a t = a 0 C 0 = C 1 = 0 W 1.11 A 0 x 0 x 1 W Wronsky Wronskian Wronskian non-zero Linear Inhomogeneous Second Order Ordinary Differential Equation with Constant Coefficients x xt t a: a 0 qt: 17 d x + ax = qt 1.13 dt 1.13 expi a t exp i a t sin a t cos a t 1 x 0 x p x p = Xx 0 X Xt 1.13 x p d x p dt + ax d X p = x 0 dt + dx 0 dx dt dt + X d x 0 dt + ax 0X = qt x d X 0 dt + x dx 0 dx 0 dt dt = x 0qt [ ] dx dt = x 0 x 0 qt dt [ ] X = x 0 x 0 qt dt dt { [ ] x p = x 0 x 0 qt dt x 0 } dt 1.14 x 0 x 1 ;

23 18 CHAPTER. superimpose 1.13 x = Cx 0 + C x 1 + x p = Cx 0 + C x 1 + x 0 { [ x 0 ] } x 0 qt dt dt 1.15 C, C : 1.13 x xt d x + ax + bdx dt dt = qt a, b : qt : 1.16 x = X exp bt X Xt = [ d X dt d x dt + ax + bdx dt bdx dt + b X + ax + b dx dt bx ] exp bt = qt d X dt + a b X = expbt qt a = b 1.17 a b α a b X 1.16 x X = CX 0 + C X 1 + X 0 { [ X0 x = exp bt [CX 0 + C X 1 ] + exp btx 0 { ] } X 0 expbt qt dt dt [ X0 ] } X 0 expbt qt dt dt 1.18a 1.18b C, C X 0, X X 0 = expiαt, X 1 = exp iαt 1.19a, b x expλt λ : 1.16 d x + ax + bdx dt dt = λ + a + bλ x = x 0 λ λ + bλ + a = 0 λ = b ± b a λ 1 = b + b a, λ = b b a 1.1a, b 18

24 a b expλ 1 t, expλ t x = C expλ 1 t + C expλ t 1. C, C : λ C = C C C 1.0 expλ 1 t expλ t 1 x 0 1 x x p = Xx 0 X Xt d x p dt + ax p + b dx p dt = x 0 d X dt = x 0 d X dt + + dx dt dx 0 dt + X d x 0 dt dx dx 0 dt + bx 0 + ax 0X + b dt = qt [ ] d expbt x dx 0 = expbt x 0 qt dt dt [ ] dx = exp bt x 0 expbt x 0 qt dt dt { [ ]} X = exp bt x 0 expbt x 0 qt dt dt dx x 0 dt + X dx 0 dt x p = Xx { x = Cx 0 + C x 1 + x 0 exp bt x 0 [ ]} expbt x 0 qt dt dt 1.3 C, C : 1.1a,b λ 1 λ λ 1 + λ = b x = C expλ 1 t + C expλ t { [ ]} + expλ 1 t exp [λ λ 1 t] exp λ t qt dt dt 1.4 t { s } x = C expλ 1 t + C expλ t + exp [λ1 t s + λ s r] qr dr ds 1.5 b a < 0 λ 1 = b + i a b β, λ = b i a b = β t { s } x = C expβt + C expβ t + exp [βt s + β s r] qr dr ds

25 0 CHAPTER. 1.6 a = b 1.0,1.1a,b λ λ = b 1 expλt = exp bt 1 x 1 = X exp bt X Xt d x 1 dt + ax 1 + b dx 1 dt d X dt = 0 X = C 1 + C t = d x 1 dt + b x 1 + b dx 1 dt = d X dt exp bt = 0 x 1 = C 1 + C t exp bt exp bt x 1 C 1 = 0, C = 1 1 t exp bt Wronskian W W = exp bt 0 [ ] a, b, c, A d x + ax + bdx dt dt = A expct x expλt λ : λ λ + bλ + a = 0 λ = b ± b a λ 1 = b + b a, λ = b b a 1.9a, b b a λ λ 1, λ x h x h = C expλ 1 t + C expλ t C, C : 1.30 b = a λ λ = b x h x h = C + C t exp bt C, C : x p x p expct c, a, b c + bc + a 0 x p x p = A expct c + bc + a 1.3 0

26 c + bc + a = 0 i a b ii a = b c + b = 0 i 1 x p = At expct c + b 1.33 ii x p = 1 At expct x h 1.3,1.33,1.34 x p 1.8 x x = x h + x p [ ] 1.8 x p x p = Bt n expct 1.35 B:, n: 1.8 n, B d x p dt + ax p + b dx p dt = c + bc + a Bt n e ct + nc + bbt n 1 e ct + nn 1Bt n e ct = A expct 1.36 c +bc+a 0 1 A expct n, B :n = 0, B = A/ c + bc + a c + bc + a = 0 a b n, B : n = 1, B = A/c + b c + bc + a = 0 a = b 3 n, B :n =, B = A/ a, b, c, A m d x + ax + bdx dt dt = Atm expct 1.37 Linear Inhomogeneous Second Order Ordinary Differential Equation x xt t, pt, qt, Rt: d x + ptx + Rtdx dt dt = qt.1 x = Xf X Xt, f exp[ Rt dt] [ f d X dt + dx df dt dt + X d f + ptxf + Rt f dx dt dt + X df ] = qt dt 1

27 CHAPTER. where P t pt + d { [ X d ] } dt + f pt + dt + Rtdf f 1 X = f 1 qt dt d X + P tx = Qt. dt [ d ] f dt + Rtdf f 1, Qt f 1 qt.3a, b dt. 1 X 0 X 0 X 0 t X 0 1 X 1 X 1 X 1 t X 0 X 1 d X 0 dt + P tx 0 = 0, d X 1 dt + P tx 1 = 0.4a, b X 1 X 0 [ d ] [ X 0 d ] X 1 X 1 + P tx dt 0 X 0 + P tx dt 1 X 1 dx 0 dt X 0 dx 1 dt d X 0 d X 1 = X 1 X dt 0 dt = d dx 0 dx 1 X 1 X 0 = 0 dt dt dt = constant A A 0.5 X 0 X X 1 = AX 0 [ ] X0 dt. X = Y X 0 Y Y t. d X dt + P tx = d Y dt X 0 + dy dt [ dy = X0 dt { X = X 0 X = X 0 { dx 0 dt + d X 0 dt Y + P ty X 0 = Qt ].6 X 0 Qt dt + constant [ ] } X 0 Qt dt + constant dt + constant [ ] } X 0 Qt dt dt + CX 0 + C X 1.7 X 0 X 0 C, C : 1, x = fx 0 { [ X0 ] } X 0 Qt dt dt + f [CX 0 + C X 1 ].8

28 3 1 d x + ax = 0.9 dt x xt a : t : t A t t B t B = t A + π/ a y dx/dt 1 dx dt y = 0, dy dt + ax = 0.10a, b.10a,b x, y vector x x x, y T T transposition vector matrix matrix A unit diagonal matrix I A [ 0 1 a 0 ], I [ a,b ].11a, b dx dt Ax = 0 0 : zero vector, 0 = 0, 0 T.1 x, y expλt λ :.1 λ I Ax = 0.13 x 0 λ I A zero λ λ I A = λ + a = 0 λ = ±i a i 1.14 λ 1 = i a, λ = i a.15a, b λ 1 vector x 1 = x 1, y 1 T x = x, y T.13 x 1 λ x 1 = C expλ 1 t, y 1 = C expλ 1 t, C, C :.16a, b λ 1 x 1 = y 1 C = λ 1 C.16c 3

29 4 CHAPTER. x ax 1 = λ 1 y 1 ac = λ 1 C.16d x = c expλ t, y = c expλ t, c, c :.17a, b λ x = y c = λ c.17c ax = λ y ac = λ c.17d x 1 = x 1, y 1 T = C 1, λ 1 T expλ 1 t x = x, y T = c 1, λ T expλ t.18a.18b.18a,b C, c 1 eigenfunction normalization t A t t B t B = t A + π/ a [ ] tb < x T 1 1, x 1 > x T 1 x 1 dt t B t A t A 1 tb = x t B t 1x 1 + y1y 1 dt A t A = CC 1 + a = 1,.19a CC = a C = C = a x 1e, x e x 1e = 1, λ 1 T expλ 1 t 1 + a,.19b.0a x e = 1, λ T expλ t 1 + a.0b.13 x = C 1 x 1e + C x e C 1, C :.1.1,.13 dx dt = Ax, Ax = λ I x = λx.a, b xt t = 0 Taylor dx xt = x0 + t + 1 d x t d n x t n +.3 dt dt 0 n! dt 0 n 0 4

30 .a t xt = x0 + Ax 0 t + 1 A dx t A dn 1 x dt n! dt 0 n 1 t n + 0 = x0 + Ax0 t + 1 AAx0 t n! An x0 t n +.4.b xt = x0 + λ I x0 t n! λ I n x0 t n + = x0 + λx0 t n! λn x0 t n + = x0 n=0 1 n! λn t n = x0 expλt a xt = x0 expλt = x0 expat.6 matrix A expa n=0 5 1 n! An.7 [ ] 3 constant vector a = a 1, a, a 3, b = b 1, b, b 3 T inner product a b a b a 1 b 1 + a b + a 3 b 3 =< a, b > a a a a vector a a b zero vector a b a b = 0 C 1, C C 1 a T + C b = 0 a, b T C 1 a a T + C a b = 0, C 1 b T a T + C b T b = 0 a b = 0 a a T 0 C 1 = 0 b T a T = a b = 0 b T b 0 C = 0 vector 5

31 6 CHAPTER. t A t t B ft, gt < f, g > < f, g > 1 t B t A tb t A ft gt dt * < f, f > f norm f = < f, f > zero f, g zero C 1, C C 1 f + C g = 0 f, g C 1 < f, f > +C < f, g >= 0, C 1 < g, f > +C < g, g >= 0 < f, g >= 0 < f, f > 0 C 1 = 0 < g, f >= 0 < g, f >=< f, g > < g, g > 0 C = 0 orthogonal function f, g 6

32 Chapter A second order ordinary differential equation for u uz in the region < z <, with two real constants α and γ: z d u + γ zdu αu = dz dz z = 0 z = 1 Assuming that the solution of Eq1.1 is expressed by a power series of z with coefficients a 0, a 1, a, : we have the following equations: thus u = a n z n = a 0 + a 1 z + a z + a 3 z 3 +, 1. n=0 du dz = n=1 a n nz n 1, d u dz = n= a n nn 1z n, z d u + γ zdu dz dz αu = a n nn 1z n 1 + γ z a n nz n 1 α a n z n n= n=1 n=0 = αa 0 + γa 1 a 1 α + 1z + a γ + 1z + a n nγ + n 1z n 1 a n α + nz n = n=3 The coefficients should be chosen so that 1. satisfies Eq1.3: n= a 1 = α γ a 0, a = α + 1 γ + 1 a 1,, a n = α + n 1 nγ + n 1 a n 1 a n = where α n and γ n stand for the following expression: α n n!γ n a 0, for n = 1,,, 1.4 α n αα + 1α + α + n 1 = Γα + n/γα, 1.5 7

33 8 CHAPTER 3. in terms of the gamma function Γx: Γx 0 exp t t x 1 dt with Re{x} > 0 for a genaral case u = a 0 u 1 a 0 u u 1 = 1 + n=1 α n z n n!γ n = n=0 α n z n n!γ n 1.7 u 1 Kummer F α, γ; z 1.1 The other solution u = Au 1 where A = Az, is a function to be determined as follows: z d u + γ zdu dz dz αu d A = z dz u 1 + da du 1 dz dz + d u 1 da dz A + γ z dz u 1 + du 1 dz A αau 1 d A = z dz u 1 + da du 1 + γ z da dz dz dz u 1 = 0 d A dz + du 1 u 1 dz + γ z da z dz = da [ dz = c du 1 1 exp u 1 dz + γ z ] 1 dz = c 1 exp[ γlnz + z] z u 1 c 1 : an integration constant u = Au 1 { } 1 Au 1 = u 1 c 1 exp[ γlnz + z] dz + c u c :an integration constant Eq1.1 u the general solution of Eq1.1: C 1, C :integration constants 1 u = u 1 exp[ γlnz + z] dz 1.10 u 1 u = C 1 u 1 + C u 1.11 ΓαΓγ α z F α, γ; z = z 1 γ expt t α 1 z t γ α 1 dt Γγ 0 8

34 Chapter 4 Vector vector tensor / /9.10./10.18./in Tex vector magnitude direction scalar vector vector vector vector a a a, etc vector vector a a a a zero vector zero-vector 0 zero-vector zero vector a vector unit vector e = a/ a nonzero-vector vector a b a b a b a b parallel a b unti-parallel vector a scalar c a = cb a b c > 0 c < 0 vector scalar c a c b a = c a c b a = c a c b a, 0a = 0 zero vector, c a + c b a = c a a + c b a vector a b a b a b normal, perpendicular vector 9

35 30 CHAPTER 4. VECTOR 3 vector a, b, c scalar 3 vector vector 3 vector e 1, e, e 3 vector a vector a = a 1 e 1 + a e + a 3 e 3 = a 1, a, a 3 a 1, a, a 3 vector a component vector e 1 = 1, 0, 0, e = 0, 1, 0, e 3 = 0, 0, 1 vector a a a a 1 + a + a 3 vector a ca = ca 1 e 1 + ca e + ca 3 e 3 c:, scalar vector matrix a = a 1 a a 3 vector, a = a 1 a a 3 vector vector a a = a 1, a, a 3 T T transposition matrix matrix determinant vector tensor a = a 1 e 1 + a e + a 3 e 3 = a 1, a, a 3 a i i = 1,, 3 vector a 1, a, a 3 1 set index index 1/4 tensor tensor vector {a} i = a i vector a, b a = a 1 e 1 + a e + a 3 e 3 = a 1, a, a 3, b = b 1 e 1 + b e + b 3 e 3 = b 1, b, b 3 vector a + b a + b = a 1 e 1 + a e + a 3 e 3 + b 1 e 1 + b e + b 3 e 3 = a 1 + b 1 e 1 + a + b e + a 3 + b 3 e 3 = a 1 + b 1, a + b, a 3 + b 3 30

36 31 vector c = a + b commutative law a + b = b + a a + b = a 1 + b 1, a + b, a 3 + b 3 = b 1 + a 1, b + a, b 3 + a 3 = b + a associative law a + b + c = a + b + c = a + b + c a, b c a a + c b b = 0 c a = c b = 0 a b 1 c a + c b > 0 1 vector c = a + b a = c b zero-vector 0 = 0, 0, 0, a vector a 0 = a a a = a 1 e 1 + a e + a 3 e 3 = a 1 e 1 a e a 3 e 3 = a 1, a, a 3 P 1 vector x 1 vector P vector x vector P 1 P vector x x 1 vector vector vector vector vector [ ] d Alembert Newton N F 1, F,, F N F k k = 1,,, N N F k = 0 k=1 N K = F k k=1 K dynamic balance vector dyad inner product, scalar product, dot product a b a b = a b cos θ θ a b [rad] a b = b a a b projection P roja b b = a b = P rojb a a 31

37 3 CHAPTER 4. VECTOR a a = a = a a = a a b = a b cos θ a b Cauchy vector a b = 0 a b normal, perpendicular a b zero-vector vector c a a + c b b = 0 a c a a + c b b = a 0 = 0 a a 0, a b = 0 c a = 0, b c a a + c b b = b 0 = 0 b b 0, b a = 0 c b = 0 a b c = b a ba/a a vector vector a/ a c/ c a b vector A a A = c a a + c c b c vector c a, c b : e 1 e 1 = 1, e e = 1, e 3 e 3 = 1, e 1 e = 0, e e 3 = 0, e 3 e 1 = 0 a a = a e 1 e 1 + a e e + a e 3 e 3 = a 1 e 1 + a e + a 3 e 3 a b a b = a 1 e 1 + a e + a 3 e 3 b 1 e 1 + b e + b 3 e 3 = a 1 e 1 b 1 e 1 + b e + b 3 e 3 + a e b 1 e 1 + b e + b 3 e 3 +a 3 e 3 b 1 e 1 + b e + b 3 e 3 = a 1 b 1 + a b + a 3 b 3 = b 1 a 1 + b a + b 3 a 3 = b a a b + c = a b + a c matrix a b = b 1 a 1 a a 3 b = a 1 b 1 + a b + a 3 b 3 = b 1 b b 3 b 3 a 1 a a 3 = b a matrix dyad tensor 3 a b = a 1 b 1 + a b + a 3 b 3 = a i b i = a i b i i=1 3

38 i=1 3 Einstein summation notation, summation convention index taking the summation for the repeated index 1 index 3 index 3 vector a b + b a = a i b i + b i a i = a i b i + b j a j = a k b k = a 1 b 1 + a b + a 3 b 3 tensor index 33 a i 1 tensor, a i b k, A ij tensor, C ijk 3 tensor i, j, k = 1,, 3 tensor a i b k dyad a i b k a k b k index a k b k 0 tensor scalar tensor δ ij ε ijk δ ij tensor, tensor, Kronecker delta ; δ ii = 3 index i j 1 δ 11 = δ = δ 33 = 1 zero index 3 Kronecker delta δ ij a j = a i ; δ ij δ jk = δ ik ; δ ij δ jm δ mn = δ in Eddington epsilon 3 tensor ε ijk i, j, k 1,, i, j, k zero Eddington epsilon 3 tensor isotropic tensor Kronecker delta tensor 4 tensor C ikmn C ikmn = aδ ik δ mn + bδ im δ kn + cδ in δ km a, b, c tensor Kronecker delta 33

39 34 CHAPTER 4. VECTOR outer product, vector product, cross product a b, a b a b = a b e sin θ θ a b [rad] e a b vector determinant a b = e 1 e e 3 a 1 a a 3 b 1 b b 3 = e 1 a b 3 a 3 b e a 1 b 3 a 3 b 1 + e 3 a 1 b a b 1 a b = b a vector zero-vector a a = 0, a b = 0 a b [ ] e 1 e e 3 e 1 e = e 3 e 1 e E 3 e 1 e = E 3 = e 3 vector vector vector vector vector x F vector moment M = x F vector vector a b a b a b = a b e sin θ a b a b sin θ = 1 a b sin θ = 0 vector e 1 e = e 3, e e 3 = e 1, e 3 e 1 = e, e e 1 = e 3, e 3 e = e 1, e 1 e 3 = e, e 1 e 1 = 0, e e = 0, e 3 e 3 = 0, a b = a 1 e 1 + a e + a 3 e 3 b 1 e 1 + b e + b 3 e 3 = a 1 e 1 b 1 e 1 + b e + b 3 e 3 +a e b 1 e 1 + b e + b 3 e 3 +a 3 e 3 b 1 e 1 + b e + b 3 e 3 = a 1 b 1 e 1 e 1 + a 1 b e 1 e + a 1 b 3 e 1 e 3 34

40 35 +a b 1 e e 1 + a b e e + a b 3 e e 3 +a 3 b 1 e 3 e 1 + a 3 b e 3 e + a 3 b 3 e 3 e 3 = a 1 b e 3 a 1 b 3 e a b 1 e 3 + a b 3 e 1 + a 3 b 1 e a 3 b e 1 = e 1 a b 3 a 3 b e a 1 b 3 a 3 b 1 + e 3 a 1 b a b 1 a b + c = a b + a c tensor {a b} i = ε ijk a j b k = ε imn a m b n ; ε ijk a j b k = ε ikj a j b k index Eddington epsilon index dyad dyad a b a b = a 1 a a 3 b1 b b 3 = a 1 b 1 a 1 b a 1 b 3 a b 1 a b a b 3 a 3 b 1 a 3 b a 3 b 3 a b trace Tra b = a 1 b 1 + a b + a 3 b 3 = a b dyad a b + c = a b + a c, a cb = ca b, a + b c = a c + b c, ca b = ca b a b b a dyad dyadic scalar 3 triple scalar-product vector a, b, c 6 a b c a b c = a 1 a a 3 b 1 b b 3 c 1 c c 3 a b c = a b c = a 1 b c 3 b 3 c a b 1 c 3 b 3 c 1 + a 3 b 1 c b c 1 a b c = b c a = c a b, a b c = a c b = b a c 35

41 36 CHAPTER 4. VECTOR scalar 3 tensor a b c = a i ε ijk b j c k = ε ijk a i b j c k = ε kmn a k b m c n vector 3 triple vector-product a b c = a b c a b c a b c a b c a b c = e 1 e e 3 a 1 a a 3 b c 3 b 3 c b 1 c 3 + b 3 c 1 b 1 c b c 1 = e 1 [a b 1 c b c 1 a 3 b 1 c 3 + b 3 c 1 ] e [a 1 b 1 c b c 1 a 3 b c 3 b 3 c ] +e 3 [a 1 b 1 c 3 + b 3 c 1 a b c 3 b 3 c ] = e 1 [b 1 a c + a 3 c 3 c 1 a b + a 3 b 3 ] e [ b a 1 c 1 + a 3 c 3 + c a 1 b 1 + a 3 b 3 ] +e 3 [b 3 a 1 c 1 + a c c 3 a 1 b 1 + a b ] = ba c ca b ba c ca b = bc a cb a = a cb a bc = c ab b ac vector 3 tensor {a b c} i = ε ijk a j ε kmn b m c n = ε ijk ε kmn a j b m c n Eddington epsilon ε ijk ε pqr = δ pqr ijk = δ ipδ jq δ kr δ ip δ jr δ kq + δ iq δ jr δ kp δ iq δ jp δ kr + δ ir δ jp δ kq δ ir δ jq δ kp Kronecker delta i, j, k p, q, r Kronecker delta p, q, r p, r, q p, q, r q, r, p q, p, r p, q, r r, p, q r, q, p k = p ε ijk ε kqr = δ kqr ijk = δqrk ijk = δqr ij = δ iq δ jr δ ir δ jq Kronecker delta index superscript index subscript {a b c} i = ε ijk a j ε kmn b m c n = δ im δ jn δ in δ jm a j b m c n 36

42 37 = a j b i c j a j b j c i = a cb i a bc i vector 3 a b c = ba c ca b tensor x = x 1, x, x 3, scalar φ φx = φx 1, x, x 3 φ x 1, x, x 3 dφ = φ x 1 dx 1 + φ x dx + φ x 3 dx 3 ; dφ = φ x k dx k φ x 1 x, x 3 dφ = φ x 1 dx 1 scalar φ φψ, ψ ψx = ψx 1, x, x 3 dφ = dφ dφ dψ = dψ dψ scalar φ φx, t = φx 1, x, x 3, t ψ x 1 dx 1 + ψ x dx + ψ x 3 dx 3 x 1 x 1 t, x x t, x 3 x 3 t φ dφ = φ φ dt + dx 1 + φ dx + φ dx 3 t x 1 x x 3 dφ dt = φ t + φ dx 1 x 1 dt + φ dx x dt + φ dx 3 x 3 dt tensor vector v vx = v 1, v, v 3 v v 1 v 1 x 1, x, x 3, v v x 1, x, x 3, v 3 v 3 x 1, x, x 3 v dv = dv 1, dv, dv 3 v tensor dv k = v k x 1 dx 1 + v k x dx + v k x 3 dx 3 for k = 1,, 3 dv k = v k x m dx m for k = 1,, 3 [ ] N Hamilton H Hq, p, t :q qt = q 1, q,, q N, :p pt = p 1, p,, p N dh = H H dt + dq 1 + H dq + + H dq N t q 1 q q N 37

43 38 CHAPTER 4. VECTOR + H p 1 dp 1 + H p dp + + H p N dp N dh dt = H t + N n=1 [ H dq n q n dt + H ] dp n p n dt gradient operator nabla x 1, x, x 3 x 1, x, = e 1 + e + e 3 x 3 x 1 x x 3 scalar φ φx 1, x, x 3 gradient φ = φ x 1, φ x, φ x 3 = e 1 φ x 1 + e φ x + e 3 φ x 3 φ grad φ tensor { φ} k = φ x k v divergence v = v 1 x 1 + v x + v 3 x 3 ; v = v k x k tensor v div v vector v v = 0 v divergence-free vector v rotation v = e 1 e e 3 / x 1 / x / x 3 v 1 v v 3 = e 1 v3 x v x 3 e v3 x 1 v 1 x 3 v + e 3 v 1 x 1 x v rot v curl v curl v rot v v = 0 v rotation-free vector irrotational vector v v vorticity v = 0 38

44 39 v dyad v = grad v grad v = e 1 v 1 e 1 + v e + v 3 e 3 + e v 1 e 1 + v e + v 3 e 3 x 1 x +e 3 v 1 e 1 + v e + v 3 e 3 x 3 v 1 v v 3 x 1 x 1 x 1 v 1 v v 3 = x x x v 1 v v 3 x 3 x 3 x 3 = v; { v} mn = v n x m grad v trace Tr v = v grad v { v} mn = v n x m = ɛ mn + Ω mn ɛ mn Ω mn ɛ mn 1 vn + v m, Ω mn 1 vn v m x m x n x m x n ɛ mn index ɛ mn = ɛ nm tensor symmetric tensor Ω mn = Ω nm tensor tensor ɛ mn trace ɛ mm = v trace 6 3 ɛ 1 = ɛ 1, ɛ 13 = ɛ 31, ɛ 3 = ɛ 3 Ω mn zero Ω 11 = Ω = Ω 33 = 0 3 Ω 1 = Ω 1, Ω 3 = Ω 3, Ω 13 = Ω 31 Ω mn { v} k = ε kmn v n x m = 1 ε kmn ε knm v n x m = ε kmn Ω mn vector vector ɛ mn + Ω mn 39

45 40 CHAPTER 4. VECTOR [ ] vector v vx = v 1, v, v 3 tensor dv n = A nm dx m with A nm v n x m tensor A nm matrix determinant Jacobian J Dv 1, v, v 3 /Dx 1, x, x 3 J Dv 1, v, v 3 Dx 1, x, x 3 = v 1 x 1 v 1 x v 1 x 3 v x 1 v x v x 3 v 3 x 1 v 3 x v 3 x 3 J 0 tensor B kn B kn dv n = B kn A nm dx m = δ km dx m = dx k dx k = B kn dv n 1 A nm dx m dv n dv n = A nm dx m affine v n = A nm x m λ A nm x m = λδ nm x m = λx n λ eigenvalue x n vector [ ] x = x 1, x, x 3 x 3 θ v = v 1, v, v 3 Jacobian J = 1 v 1 = x 1 cos θ + x sin θ, v = x 1 sin θ + x cos θ, v 3 = x 3 cos θ sin θ 0 J = sin θ cos θ 0 = vector x = x 1, x, x 3 x = r 1, r, r 3 = r r r r = r = x 1 + x + x 3 scalar φ φx = φx 1, x, x 3, etc. vector v vx = vx 1, x, x 3 = v 1, v, v 3, etc. 40

46 41 1 grad φψ = φψ = ψ φ + φ ψ = ψ φ + φ ψ grad r = r = e x unit vector r x 1, r x, r = x x 3 r = e 1 grad = grad r d 1 = x r dr r r = e 3 r φ = c/r c: Coulomb div φv = φv = v φ + φ v div x = x = 3 x div = 1 r r div x x x = r 3 r x div = 1 x 1 div x 3x = 0 div grad r 3 r3 r 5 r = 0 3 rot φv = φv = φ v + φ v = φ v v φ rot x = x = 0; x n { x} k = ε kmn = ε kmn δ mn = ε kmm = 0 x m x x rot = = 1 1 r r r x x = 1 r r x x = div grad φ = φ div grad φ = φ = φ + φ + φ x 1 x x 3 φ = φ = φ Laplace Laplacian φ = c/r c: 3 φ = 0 grad φ grad φ = φ = φ = 0 5 rot grad φ = φ = 0 subscript = 1 = 1 φ = 1 φ 41

47 4 CHAPTER 4. VECTOR subscript 1 φ + 1 φ = φ = 0 6 div rot v = v = 0 subscript v = 1 v scalar 3 v = 0 v = 0 1 v = 1 v = v 7 u v = v u u v 8 u v = u v + v u v u u v 9 u v = u v v = v u v u v v = v u v u v v u = v u u = u u v v u u = u u v v u u v = u u v + v u v = u v + u v + v u + v u u u operator, v v operator 10 v v = [v v + v v] 11 v = v v ; rot rot v = grad div v div grad v 1 v = v = [ v v] = v v = 0 div rot rot v = div grad div v div grad div v = 0 13 v = v v = v rot rot rot v = grad div rot v div grad rot v = div grad rot v 4

48 43 scalar potential φ vector potential A v = 0 v scalar potential φ v = φ v = φ = 0 v = 0 v vector potential A v = A v = A = 0 vector v v = φ + A v divergence rotation φ A v φ = φ = 0 φ = φ = fx A = A = 0 A = A = fx u t = u κ x u t = κ u u t c u x = 0 u t c u = 0 Laplace Poisson fx Laplace Poisson, fx 1 1 to be continued 43

49 Chapter x, y, z vector e 1, e, e 3 vector r r xe 1 + ye + ze 3 1 r parameter u, v x xu, v, y yu, v, z zu, v v u 1 u-curve u v 1 v-curve u, v 1 surface r dr = dr = dxe 1 + dye + dze 3 x x y du + u v dv y e 1 + du + u v dv e + z z du + u v dv e 3 = r r du + dv 3 u v ds ds = dr dr = r du + r u u r v dudv + r dv 4a v ds = Edu + F dudv + Gdv 4b E, F, G E r, F r u u r v, G r 5a, b, c v 4b r ru, v 1 1 r/ u ucurve vector r/ v v-curve vector u, v r/ u r/ v = 0 u-curves u = constant v-curves v = constant 44

50 vector vector N vector n N r u r v, 45 n r u r 1 6a, b v N N 3 vector t t dr ds = r du u ds + r dv v ds 7 t s vector n κ n = r d u u ds + r u κ n n n = κ n = n dt ds = d r ds = κ nn du + r ds u v κ n = e r u du ds du ds du ds dv ds + r d v v ds + r v + r u v + f du dv ds ds + g κ n = edu + fdudv + gdv ds κ n = edu + fdudv + gdv Edu + F dudv + Gdv du dv ds ds + r v dv 8 ds dv ds dv 9a ds e n r u, f n r u v, g n r 10a, b, c v 9b edu + fdudv + gdv r = ru, v 9b 9c κ n E edu + κ n F fdudv + κ n G gdv = 0 11 Ef F edu + Eg Gedudv + F g Gfdv =

51 Chapter /revised and enlarged April Introduction d x dt = f x, dx x xt, t : 1.1 dt fx, dx/dt x dx/dt t explicit autonomous system 1.1 dy dt = fx, y, y dx dt 1.a, b x, y 1 N N 1 1.a,b 1 dy dt = fx, y, dx dt = gx, y 1.3a, b gx, y x y x, y gx, y 0 1.3a,b dy dx = fx, y gx, y x y = yx x, y fx, y 0 dx/dy = gx, y/fx, y x = xy t parameter a,b 1.3a,b xt, yt 1.4 yx x y, phase space a,b

52 a,b x, y fx, y = 0 gx, y = 0 x = X, y = Y X, Y = constant fx, Y = 0, gx, Y = 0 1.5a, b X, Y 1.3a,b equilibrium point x xt, ỹ ỹt x = X + x, y = Y + ỹ fx, y, gx, y x, ỹ Taylor 1 where f a x X,Y fx + x, Y + ỹ = fx, Y + a x + bỹ = a x + bỹ, gx + x, Y + ỹ = gx, Y + c x + dỹ = c x + dỹ,, b f, c y X,Y g, d x X,Y g y X,Y 1.6a 1.6b 1.7a, b, c, d 1.3a,b dỹ dt = a x + bỹ, d x dt = c x + dỹ 1.8a, b x, ỹ expλt λ λ b + cλ + bc ad = λ unstable stable zero neutral 1.8a,b dỹ d x = a x + bỹ c x + dỹ dy px, y = dx qx, y.1 47

53 48 CHAPTER 6. p0, 0 = 0, q0, 0 = 0 px, y, qx, y x = 0, y = 0 regular 0,0 singular point Poincaré dy dx = ax + by cx + dy. a, b, c, d ad bc 0. Poincaré a nodal point b saddle point c center point d asymptotically spiral point [1] b c + 4ad > 0 ad bc < 0 [] b c + 4ad > 0 ad bc > 0 [3] b c + 4ad < 0 b + c = 0 [4] b c + 4ad < 0 b + c 0 [5] b c + 4ad = 0 a [5] dy dx = y x y = Cx C.3a b dy dx = x y x y = C C = 0 c dy dx = x y x + y = C.3b.3c 48

54 d x/dt + x = 0 x = sint y = cost d dy dx = x + y x y x Lnx + y = tan 1 y + C.3d x = r cos θ, y = r sin θ r = C expθ 49

55 50 CHAPTER 6. [ ] x xt t: [1] d x dt + x = 0, [] d x dt x = 0, [3] d x dt + dx dt + x = 0, [4] d x + sinx = 0, dt [5] d x dt + x x = 0 Satellite equation, [6] d x dt + x + x3 = 0 Duffing equation, d x [7] dt + x = 1 x dx Van del Pol equation, dt d x [8] dt + x = dx dt 1 3 dx Rayleigh equation 3 dt 3 x f fx x x 0 A A: Taylor fx = a 0 + a 1 x x 0 + a x x a n x x 0 n + = fx 0 + [Df]x x 0 + [D f] x x [Dn f] x x 0 n +, 3.1 n! d where [D n n f f] = for n = 0, 1,, dx n x 0 fx x x 0 A regular analytic x y fx, y x x 0 A, y y 0 B A, B: Taylor fx, y = a 00 + a 10 x x 0 + a 01 y y 0 + a 0 x x 0 +a 11 x x 0 y y 0 + a 0 y y 0 + = fx 0, y 0 + [D X f] x x 0 + [D Y f] y y 0 50

56 {[ D X f ] x x 0 + [D X D Y f] x x 0 y y 0 + [ D Y f ] y y 0 } n! { n m=0 [ nc m D n m X D m Y f ] } x x 0 n m y y 0 m +, 3. n! where nc m = m n, m!n m! [D n n f Xf] = and [D n n f x n Y f] = for n = 0, 1,, x 0,y 0 y n x 0,y 0 fx, y x x 0 A, y y 0 B x y y yx fx, y x yx Taylor yx = a 0 + a 1 x x 0 + a x x a n x x 0 n where a n 1 n! d n y dx n a 0 = yx 0 = y 0 y y 0 x 0 for n = 0, 1,, y y 0 = a 1 x x 0 + a x x a n x x 0 n x, y fx, y Taylor fx, y = n=0 { 1 n [ nc m D n m X D m Y f ] } x x 0 n m y y 0 m n! m=0 y y x, yx fx, y Taylor 3.5 [ ] x y y yx fx, y x x 0 A, y y 0 B 1 dy dx = fx, y 3.6 x = x 0 y = y 0 y 0 = yx 0 x 0 y = yx // [ ] 3.6 y = yx 3.6 x = x 0 y 0 = yx 0 0! a 0, dy = fx 0, y 0 = a 00 1! a 1, 3.7a, b dx x 0 51

57 5 CHAPTER 6. d y dx d 3 y dx 3 = x 0 = = x 0 [ df f = dx x 0 x + dy ] [ f f = dx y x 0,y 0 x + f f ] y x 0,y 0 d f dx x 0 = = a 10 + a 00 a 01! a, 3.7c [ d f + d y f dx x dx y + dy ] d f dx dx y f x + f f x y + f f y + f x f y + f f y x 0,y 0 x 0,y 0 = a 0 + a 00 a 11 + a 00a 0 + a 10 a 01 + a 00 a 01 3! a 3, 3.7d a n 1 d n y = 1 d n 1 f = 1 n! dx n x 0 n! dx n 1 x 0 n! x + f n 1 f y for n = 1,, 3, x 0,y 0 3.7e yx yx = a 0 + a 1 x x 0 + a x x a n x x 0 n a n for n = 0, 1,, x 0, y 0 = yx 0 fx, y x // 1 1 [ ] L t = 0 m x x xt d Alembert m d x = kx L + mg 3.9 dt k g dx x = L, = 0, at t = dt 5

58 equilibrium solution x = x 0 = constant x 0 = L + mg k X = x x 0 X Xt 3.9 m d X dt = kx Xt = a sinωt + b cosωt, where ω k m, a, b : xt xt = a sinωt + b cosωt + x a, b x0 = b + x 0 = L b = L x 0 = mg k, dx = aω = 0 a = 0 dt xt xt = x 0 mg k mg cosωt = L + [1 cosωt] 3.11 k x 0 harmonic oscillation mg/k ω = k/m 3.11 xt t = 0 Taylor xt = x0 + dx t + 1 dt n! d x dt t d 3 x dt 3 t d 4 x dt 4 t 4 0 d n x t n +, 3.1 dt n 0 0 t = ,

59 54 CHAPTER a t 3.13b 3.13c,d,e,f d x dt = k d x L + g, m 3 x dt 3 = k m dx, 3.13a, b dt d 4 x dt 4 = k m d x dt, d 5 x dt 5 = k m d 3 x dt 3 = k dx, 3.13c, d m dt d n x dt n = k m d n x dt n dn x dt n = kn m n [ x L mg k ] for n = 1,,, 3.13e d n+1 x dt n+1 = k m d n 1 x dt n 1 = kn m n dx dt for n = 1,, 3.13f 3.13a-f t = xt Taylor 3.1 xt = L + 1 d x t + 1 d 3 x t d 4 x t 4 + dt 6 dt dt = L + mg k = L mg k n=1 k n t n n!m n [ kt m k t 4 4m + k3 t 6 ] 70m + 3 Ot Taylor asymptotic series D [ ] y yx [1] dy dx = y, [] d y dx y = 0 54

60 [ ] x xt t: d x dt = F x, dx dt, t 4.1 F x, dx/dt, t t T F x, dx/dt, t = F x, dx/dt, t + T 4.1 T x xt = xt + T x0 = xt, dx = dt 0 dx dt T 4.a, b // [ ] xt = xt + T 4.a,b t = 0 x = x 0, dx/dt = v x = X X Xt X d X dt = F X, dx dt, t t = T + s F x, dx/dt, t d XT + s dxt + s = F XT + s,, T + s ds ds dxt + s = F XT + s,, s ds xs = XT + s s = 0 x0 = XT, XT = X0 = x 0, dx = ds 0 dxt + s ds 55 0 dxt + s = ds dxt dt a, b = v 0 4.6a, b

61 56 CHAPTER 6. XT + s = Xs 4.7 x = Xt T // 4.1 x = Xt F x, dx/dt, t x = Xt + T 4.1 Xt = Xt + T x = Xt T 4.6a,b x 0, v 0, T [ ] d x dt + x = α cosωt α, ω : parameters x = A, dx/dt = 0 at t = 0 ω 1 x = A α cost + 1 ω α 1 ω cosωt ω A = α/1 ω T = π/ω // [ ] ω = 1 x x = αt sint + A cost : xt = xt + π // ω [ ] x xt t: d x dt + x = αf x, dx dt, ωt α, ω : parameters 4.8 F x, dx/dt, ωt t <, x <, dx/dt < t, x, dx/dt t T = π/ω F x, dx, ωt + π dt = F x, dx dt, ωt 4.9 t = δ x = A, dx/dt = 0 T = π/ω x = Xt 1 δ, A, ω, α 56

62 4. 57 A = φ 1 δ, α, ω = ψ 1 δ, α 4.10a, b A = φ ω, α, δ = ψ ω, α 4.11a, b ω = φ 3 A, α, δ = ψ 3 A, α // 4.1a, b [ ] t = δ x = A, dx/dt = x = Xt x = Xt + δ t = 0 Xδ = A, dx/dt = 0 d X dt + X = αf X, dx, ωt + ωδ dt t = 0 X = A, dx/dt = 0 Xt = Xt; A, δ, ω, α 4.13 T XT ; A, δ, ω, α = A, [ ] d Xt; A, δ, ω, α dt T = a, b 4.14a,b T 4.10a-4.1b // δ parameter [ ] x y fx, y f X x, y f/ x, f Y x, y f/ y x 0, y 0 fx 0, y 0 = 0 f X x 0, y 0 f Y x 0, y 0 zere y f Y x 0, y 0 0 fx, y = y x y = F x 1 y = F x x 0 x 1 x x x fx, F x = 0, 4.16a y 0 = F x 0, b

63 58 CHAPTER 6. 3 dy dx = f Xx, y. // 4.16c f Y x, y 3 7 pp [ parameter ] x xt t: d x dt + ω x = αf x, dx dt α, ω : parameters 4.17 F x, dx/dt x <, dx/dt < x, dx/dt t = δ x = A, dx/dt = 0 T x = Xt δ, A, ω, α ω ω = φa, δ, α 4.18 α ω α 4.17 perturbation method Lindstedt 188 Poincaré 189 Lindstedt-Poincaré [ ] t = δ x = A, dx/dt = x = Xt; α, δ, A, ω 4.19 T XT ; α, δ, A, ω = A, d XT ; α, δ, A, ω = 0 4.0a, b dt T 4.18 // [ ] Duffing equation: d x dt + ω x + αx 3 = 0 x xt, t :, 0 < α 1, ω : parameter x = A, dx/dt = 0 at t = 0 58

64 4. 59 x parameter ω α x ω α x = x 0 + αx 1 + α x +, ω = ω 0 + αω 1 + α ω + x n x n t for n = 0, 1,, Duffing α d [ x dt + ω x + αx 3 = d x 0 d ] dt + x 1 ω 0x 0 + α dt + ω 0x 1 + ω 0 ω 1 x 0 + x 3 0 [ d +α x dt + ω 0x + ω 0 ω 1 x 1 + ] ω1 + ω 0 ω x0 + 3x 0x 1 + = 0 α zero x 0, x 1, x, d x 0 dt + ω 0x 0 = 0, d x 1 dt + ω 0x 1 = ω 0 ω 1 x 0 x 3 0, d x dt + ω 0x = ω 0 ω 1 x 1 ω 1 + ω 0 ω x0 3x 0x 1, x 0 x 0 x 0 = A cosω 0 t 4.1 x 1 = A d x 1 dt + ω 0x 1 = ω 0 ω 1 x 0 x 3 0 ω 0 ω 1 + 3A 4 cosω 0 t A3 4 cos3ω 0t cosω 0 t operator t sinω 0 t cosω 0 t zero solvability condition ω 1 ω 1 = 3A 8ω 0 x 1 d x 1 dt + ω 0x 1 = A3 4 cos3ω 0t 59

65 60 CHAPTER 6. : x 1 = 0, dx 1 /dt = 0 at t = 0 x 1 x 1 = A3 3ω 0 [cos3ω 0 t cosω 0 t] 4. x 1 x 0 x d x dt + ω 0x = ω 0 ω 1 x 1 ω 1 + ω 0 ω x0 3x 0x 1 = A3 ω 1 16ω 0 [cos3ω 0 t cosω 0 t] ω 1 + ω 0 ω A cosω0 t 3A5 cos ω 3ω0 0 t [cos3ω 0 t cosω 0 t] = A3 ω 1 16ω 0 [cos3ω 0 t cosω 0 t] ω 1 + ω 0 ω A cosω0 t 3A5 [cos5ω 18ω0 0 t + cos3ω 0 t cosω 0 t] cosω 0 t ω ω 0 ω = ω 1 + A ω 1 16ω 0 + 3A4 64ω 0 ω = 15A4 56ω 3 0 x = 15A4 18ω 0 d x dt + ω 0x = 3A5 cos5ω 18ω0 0 t : x = 0, dx /dt = 0 at t = 0 x x = A5 104ω 4 0 [cos5ω 0 t cosω 0 t] 4.3 x x = x 0 + αx 1 + α x + x x 1 4. x 4.3 ω ω = ω 0 + αω 1 + α ω + = ω 0 α 3A 8ω 0 α 15A4 56ω 3 0 T = π/ω A π/ω

66 4. 61 parameter α xt [ ] 1 d x dt + αdx + x = 0 0 < α 1, dt d x dt + x + αx = 0 0 < α 1 Satellite equation 61

67 Chapter 7 Lotka-Voltera [ ] x xt, y yt t: dx dt = x xy = x1 y, dy dt = y + xy = x 1y 1a, b x0 = a, y0 = b 1 x, y t > 0 1 1a x = 0 y = 1 x = 0 1b y = 0 y = 1 x = 1 0,0 1,1 u ut, v vt x = 0 + u, y = 0 + v a, b x = 1 + u, y = 1 + v 3a, b 0,0 a,b 1a,b du dt = u uv u u = C 1 expt 4a dv dt = v + uv v v = C exp t 4b C 1, C x u y v 0,0 saddle point x, y 1a,b x = 0 1a,b dx/dt = 0, dy/dt = y x = 0 1a,b y = 0 1a,b dx/dt = x, dy/dt = 0 y = 0 1a,b x, y 6

68 1 x, y t > 0 1 1,1 3a,b 1a,b du dt = 1 + u[1 1 + v] v, dv dt = u1 + v u 63 u = A 1 expit + A exp it, v = i[a 1 expit A exp it] 5a 5b A 1, A = A 1 i 1 5a,b u + v = constant 1,1 center point xt, yt dt xt + dt, yt + dt Taylor dx xt + dt = xt + dt + 1 d x dt + dt dt yt + dt = yt + dy dt + 1 dt d y dt + dt 6a,b 1a,b 3 1a,b t 1a,b xt + dt dx xt + dt = xt + dt + 1 d x dt + dt dt = xt + x1 ydt + 1 [ ] dx 1 y xdy dt + dt dt = xt + x1 ydt + 1 [ x1 y x 1xy ] dt + xt, yt 6a,b theory of dynamical system dx/dt, dy/dt flow 1a,b 1a,b 6a 6b dx = x1 y dt, dy = x 1y dt 7a, b 63

69 64 CHAPTER 7. x, y dx, dy D 4 0,0,1,1 7a,b x = 0, x = 1, y = 0, y = 1 7a,b 1 y = x 7a,b vector 1a,b 1 y y dy dt = dx dy dt dx = x 1y dy dx = x 1y x1 y dy = x 1 dx Lny y = x Lnx + constant x xy = C expx + y 8 C x0 = a, y0 = b C = ab/ expa + b > 0 1 x > 0, y > 0 xy > 0 x + y > 0 4, ,4,5 3. dx = ax bx dt 4. 1 a, b, c dx dt = ax bxy, dy dt = cy + bxy x c y a exp[ bx + y] = 3 [ ] Lotka-Voltera xt yt a by bx = 1/c 5. 1 dx dt = a bxy, 64 dy dt = cy + bxy

70 a, b, c x, y 65 [ ] xt yt xt a x y bxy cy 65

71 Chapter 8 Fourier 1 Fourier inner product a x b zero x f fx g gx f g < f, g > b a f g dx < f, g > 0 f g vector g = f < f, f > 0 fx < f, f >= 0 f 0 zero fx < f, f >> 0 f 0 f 0 / < f, f > < f 0, f 0 >= 1 normalization a x b {f n } f n f n x, n = 0, 1,,, N < f m, f n >= 0 for m n, < f m, f n >= 1 for m = n {f n } gx N gx = a n f n n=0 N J = a n < g, g > n=0 a n =< g, g > n=0 Parseval complete Parseval [ ] 0 x π sinnx n = 1,, π sinmx sinnx dx = 1 0 π 0 {cos[m nx] cos[m + nx]} dx 66

72 . FOURIER 67 = π for m = n = 0 for m n [ ] cosnx Fourier a x b x gx {cosnkx, sinnkx} n = 1,, k = π/b a gx A 0 A 0 1 b gx dx =< 1, g > b a a {cosnkx, sinnkx} gx b a gx = A 0 + [A n cosnkx + B n sinnkx] n=1 cosmkx cosnkx dx =< cosmkx, cosnkx > = b a for m = n, or = 0 for m n < sinmkx, cosnkx >= 0 for all m, n < sinmkx, sinnkx >= b a for m = n, or = 0 for m n < cosmkx, gx > { } = cosmkx, A 0 + [A n cosnkx + B n sinnkx] n=1 =< cosmkx, A m cosmkx >= A m b a A m 67

73 Chapter 9 = data = / / t data Fourier spectrum analysis data t data data Fourier 1 data Fourier data t t t data N + 1 data t j j t + t 0, f j ft j for j = 0, 1,,, N 1.1a, b t 0 t j f j data t j < t < t j+1 for j = 0, 1,,, N 1 T = N t F F t Fourier F t = [A m expiω m t + c.c.] + A M ω m π m=1 T m A m A m ω m ω m m mode 1. c.c. A m A m i 1 A M t 0 t t 0 + T F t 68

74 1. DATA FOURIER 69 Fourier 1. m M M = N t = t j [ expiω N t j = exp i π ] [ T Nj t + t 0 = exp iπ j + t ] 0 = exp iπ t 0 t t t j j = 0, 1,,, N ; data noise level M > N M = N/ M 1.3 ω M t π M N + 1 data A M, A m, A m for m = 1,,, M M +1 M +1 = N +1 M = N/ M N/ F t A M Fourier A m Fourier data {t j, f j } data {t j, f j } Fourier f j M f j = [A m expiω m t j + c.c.] + A M 1.5 m=1 1.5 t/ j = 1,,, N 1.5 t/ j = 0, 1,,, N 1 A M N N 1 t f j j=1 + t N f j j=0 = f j + f j 1 t j=1 [ N M ] [ t N 1 = A m expiω m t j + c.c. j=1 m=1 + M ] t A m expiω m t j + c.c. j=0 m=1 + A MN t { t0 +T M } = [A m expiω m t + c.c.] dt + A M T + ɛ 0 = A M T + ɛ 0 t 0 m=1 A M = 1 N f j + f j 1 t T j=1 ɛ 0 1.6a ɛ 0 ɛ 0 t, N, M f j T f 0 = f N 1.6a A M = 1 T N f j t ɛ 0 j= b

75 70 CHAPTER exp iω k t j t/ for each k; k = 1,,, M j = 1,,, N j = 0, 1,,, N 1 A k = N f j exp iω k t j t N 1 j=1 + f j exp iω k t j t j=0 { N M } = [A m expiω m t j + c.c.] + A M exp iω k t j t j=1 m=1 + N 1 j=0 { t0 +T M t 0 { M } [A m expiω m t j + c.c.] + A M exp iω k t j t m=1 } [A m expiω m t + c.c.] + A M exp iω k t dt + ɛ k = A k T + ɛ k m=1 A k = 1 T N [f j exp iω k t j + f j 1 exp iω k t j 1 ] t j=1 ɛ k for k = 1,,, M 1.7a ɛ k ɛ k t, N, M A k 1.7a T f 0 = f N 1.7a A k = 1 T N f j exp iω k t j t ɛ k for k = 1,,, M 1.7b j=1 Fourier data f ft M ft = [A m expiω m t + c.c.] + A M 1.8 m=1 ft spectrum [ ] algorithm t = t j 1.5 g j M g j [A m expiω m t j + c.c.] + A M 1.9 m=1 g j f j δ N δ g j f j 1.10 j=1 70 f j

76 . DATA 71 δ A M, A k k = 1,,, M δ A M, A k δ δ = N N g j f j g j g j gj = g j f j = 1 A M A M A M A M A M j=1 j=1 N = g j f j = a j=1 A M δ = N N g j f j g j g j = g j f j A k A k A k A k j=1 j=1 N = g j f j expiω k t j = 0 for k = 1,,, M 1.11b j=1 A k, A k j = 1,,, N 1.6a,1.7a j = 0, 1,,, N f 0, f N data data data {t j, f j } Fourier 0 t 1 ft = sinπt.1 N t = 1/N data {t j, f j } {t j, f j } = {j t, sinπj t} for j = 0, 1,,, N. data program 1 [1] M = 3, N = 10, [] M = 3, N = 0, [3] M = 5, N = 0 data computer 16 Fourier sinω 1 t Im{A 1 } zero ɛ k = 0 for k = 0, 1,, 3, 4, 5 71

77 7 CHAPTER 9. ft = sinπt /π ft = ft + T data 3 ft = sin[π t].4 T = 1/ ft T = 1 Fourier A M A k ft ft π ft = C sin T t [ ft = C + δc sin C + δc sin [ π t + δt T [ π π C + δc sin T t cos π +C + δc cos T t sin ] π T + δt t + δt ] T [ π T 1 δt T δt δtt ] t δt δtt t ] δc δt t δt data δc δc, δt, δt δt T data δt = 0 ft.5.6 ft = ft + T.7 1 data {t j, f j }.7 f j = f j+n N ft j = sinπt j f 0 = f k k k = N/ k = N k = N/.7 data ft j Df j df = 1 dt t j t f j+1 f j + O t.8a 7

78 . DATA 73 D f d f = 1 j dt t j t f j+ f j+1 + f j + O t.8b 4 π π f j = sin T t j = sin N t t j j = 0, 1,,, N,, N data N.7,.8a,b f 0 f k < t, f f 1 + f 0 t f 1 f 0 t f k+1 f k t f k+ f k+1 + f k t.9 < t.10a, b < t.10c.10a k k.10b,c T = k t data.9.10b,c data t.8a,b O t.10b,c O t ft ft + T.8a,b.10b,c noise t.10b,c noise tj+ t j tj+1 t j ft dt = t f j+1 + f j + O t.11a ft dt = t f j+ + f j+1 + f j + O t.11b.7 j = 0 j = k f 0 f k < t,.1a f 1 + f 0 t f k+1 + f k t < t.1b f k+ + f k+1 + f k t < t.1c f + f 1 + f 0 t.1a k k.1b,c T = k t.11a,b O t.1b,c O t 73

79 74 CHAPTER 9. data [ ] 1 mode mode A 1, A ω 1, ω mode ft = A 1 sinω 1 t + A sinω t.13 ω 1 ω data data Fourier ω 1 ω ft = ft + T T data N data M ω 1 /ω / / for j = 0, 1,,, N,, N

80 Chapter 10 Fourier /8.8. [ ] x N fx N fx = A n x n n=0 A n : A N x 1 fx Fourier fx Fourier fx = a 0 + [a m cosω m x + b m sinω m x] m=1 ω m = mπ m = 1,,, 1 N A n x n = a 0 + [a m cosmπx + b m sinmπx] 3 n=0 m=1 1 x 1 fx a 0 1 N { } 1 A n x n dx = a 0 + [a m cosmπx + b m sinmπx] dx 1 n=0 1 m=1 a 0 = N n=0 A n n + 1 x 1 x 1 zero n 3 coskπx k = 1,,, Fourier a k { 1 } [a m cosmπx + b m sinmπx] coskπx dx = a k 1 m=1 [ 1 N ] N = A n x n 1 a 0 coskπx dx = A n x n coskπx dx 1 n=0 n= 1 N 1 a k = A n x n coskπx dx 5 n=

81 76 CHAPTER 10. FOURIER 3 sinkπx k = 1,,, Fourier b k { 1 } [a m cosmπx + b m sinmπx] sinkπx dx = b k 1 m=1 [ 1 N ] N = A n x n 1 a 0 sinkπx dx = A n x n sinkπx dx 1 n=0 n=1 1 N 1 b k = A n x n sinkπx dx 6 n=1 1 n 5,6 cosy x 1 1 cosy x dx = Y siny 7 Y parameter Y = kπ 7 Y Y 1 1 cosy x dx = 1 1 x siny x dx = Y [ Y siny ] [ ] x n cosy x dx = 1 n/ n Y n Y siny [ ] x n siny x dx = 1 n+1/ n Y n Y siny n : 8a n : 8b a k, b k k = 1,,, a k = b k = N n=1 N n= A n 1 n/ { n Y n A n 1 n+1/ { n Y n [ 1 Y siny ] } 9a,b = m=0 [ 1 Y siny ] } Y =kπ Y =kπ n [ ] 1 n Y n Y siny m n m 1 = nc m [siny ] m=0 Y m Y n m Y { n nc m 1 m/ siny 1 n m n m! Y n+m 1 m : nc m 1 m+3/ cosy 1 n m n m! Y n+m 1 m : Y = kπ { n [ ] } 1 Y siny 9a 9b Y n Y =kπ 76

82 = n m=1 nc m 1 m+3/ coskπ 1 n m n m! kπ n+m 1 = n m=1 10 9a,b a k = b k = N n= N n=1 77 n! m! 1n m+3/ coskπkπ n+m 1 10 A n [ n m=1 A n [ n m=1 ] n! m! 13n m+3/ kπ n+m 1 coskπ ] n! m! 13n m+4/ kπ n+m 1 coskπ 11a,b,4 fx Fourier 11a 11b [ ] [1] 1 x 1 x 3 Fourier [] 1 x 1 x 4 Fourier [3] c x c c : c > 0 N Fourier N fx = B n x n n=0 [1] 1 x 1 x 3 Fourier N = 3, A 0 = A 1 = A = 0, A 3 = 1 Fourier n=1 m=1 a 0 = N n=0 A n n + 1 = 0 [ N n ] n! a k = A n n= m=1 m! 13n m+3/ kπ n+m 1 coskπ = 0 [ N n ] n! b k = A n m! 13n m+4/ kπ n+m 1 coskπ [ 3 ] 3! = m! 11 m/ kπ 4+m coskπ m=1 = 1 kπ [ 1 kπ 1 ] coskπ 3! [] 1 x 1 x 4 Fourier 77

83 78 CHAPTER 10. FOURIER N = 4, A 0 = A 1 = A = A 3 = 0, A 4 = 1 Fourier N A n a 0 = n=0 n + 1 = 1 5 [ N n ] n! a k = A n n= m=1 m! 13n m+3/ kπ n+m 1 coskπ [ 4 ] 4! = m=1 m! 1 m+3/ kπ 5+m coskπ = 48 [ 1 kπ kπ 1 ] coskπ 3! [ N n ] n! b k = A n m! 13n m+4/ kπ n+m 1 coskπ = 0 n=1 m=1 [3] c x c c : c > 0 N N fx = B n x n n=0 Fourier y = x/c, A n = B n c n fx y N N fx = B n x n = A n y n = fy n=0 n=0 Fourier a k = b k = N n= N n=1 A n [ n m=1 A n [ n m=1 a 0 = N n=0 A n n + 1 ] n! m! 13n m+3/ kπ n+m 1 coskπ ] n! m! 13n m+4/ kπ n+m 1 coskπ Fourier fy = a 0 + [a m cosmπy + b m sinmπy] m=1 78

84 Chapter 11 line of swiftest descent m g = [m/sec ] P 0 P 1 x y x, y P 0 P 1 x 1, y 1 x 1 > 0, y 1 > 0 x P, y P x P x P t, y P y P t, t : energy E E = m dxp dyp + mgx P = constant 1.1 dt dt t = 0 P 0 zero E = 0 t > 0 q qt q > 0 dxp dyp q = + = gx P 1. dt dt y yx ds q ds = qdt y y P = yx P ds = 1 + dyp dx P dx P 1.3 P 0 P 1 T x1 1 dyp T = 1 + dx P = 1 x1 f dx P q dx P g 0 f f f x P, dy P dx P = 1 xp 79 dyp 1 + dx P 1.5

85 80 CHAPTER 11. LINE OF SWIFTEST DESCENT T T 1 zero 1.4 f Euler d dx P f P P = f y P where P P dy P dx P f/ y P = f = 1 dy P P P xp dx P 1 + dyp dx P 1/ = constant = 1 A A : A dy P = dyp x P 1 + dx P dx P dy P xp = ± 1.7 dx P A x P y P = yx P 1.7 0,0 x P, y P xp 0 dy xp P xp dx P = dx dx P 0 A P 1.8 x P x P = A 1 cos θ y P = A θ sin θ ,1.10 line of swiftest descent x 1, y 1 1.9,1.10 θ 1 x 1 = A 1 cos θ 1 y 1 = A θ 1 sin θ a, b θ 1 θ 1 sin θ 1 = y 1 x 1 1 cos θ θ 1 = 0 θ 1 > a 1.11b A θ θ 1 θ θ = y [ ] x 1 θ

86 θ 1 θ 1 3 y 1 x 1 81 = 0 θ 1 = 0, θ 1 = 3 y 1 x a, b y 1 /x 1 1 θ 1 1 θ 1 = 3y 1 /x 1 y 1 /x 1 1 θ 1 = α + β α π β β β α + β β 1 6 β3 + = y [ ] x 1 β + β = x 1 y 1 α β = ± x 1 y 1 α 1.14 [ ] 1.9, b,1.14 [ ] P 0 0, 0 P 1 x 1, y 1 L y yx L = x1 0 dy 1 + dx dx [ ] P 1 x 1, y 1 P x, y x 1 < x, y 1 > 0, y > 0 y yx x S yx x dy S = π y 1 + dx x 1 dx [ ] Fermat [3 ] f fy, P = y 1 + P y yx, P dy dx S x S = π fy, P dx x 1 S yx f Euler Euler d f = f dx P y 81

87 8 CHAPTER 11. LINE OF SWIFTEST DESCENT d dx f P = yp 1 + P 1/ f = P + y dp P dx P + y dp dx f y = 1 + P 1 + P 1/ yp dp dx 1 + P 3/ 1 + P yp dp dx = 1 + P y dp dx = 1 + P yp dp dy = 1 + P Euler P dp 1 + P = dy 1 + P y = ay a : P = dy dx = ± a y 1 ay = coshz dy dx = dz sinhz dx a = a y 1 = sinhz dz = adx z = ax + b b : ay = coshz = coshax + b Euler catenary catenoid y = y 1 at x = x 1 y = y at x = x ay 1 = coshax 1 + b = coshax 1 coshb + sinhax 1 sinhb ay = coshax + b = coshax coshb + sinhax sinhb coshb sinhb coshb = ay 1 sinhax ay sinhax 1 sinh[ax x 1 ] sinhb = ay coshax 1 ay 1 coshax sinh[ax x 1 ] cosh b sinh b = 1 [ay 1 sinhax ay sinhax 1 ] [ay coshax 1 ay 1 coshax ] = sinh [ax x 1 ] 8

88 83 ay 1 ay + ay 1 ay cosh[ax x 1 ] = sinh [ax x 1 ] Z = cosh[ax x 1 ] Z Z 1 Z Z ay 1 ay Z + ay 1 + ay 1 = 0 3 ax x 1, ay 1, ay Z = ay 1 ay ± ay 1 ay ay 1 ay + 1 cosh[ax x 1 ] = ay 1 ay ± [ay 1 1][ay 1] x 1 = c c :, c > 0 x = c y 1 = y = A A :, A > 0 u ac B A/c coshu = ub ± [ub 1] coshu = 1 u = 0 coshu = ub 1 cosh u = ub coshu = ub B = 1 u coshu coshu/u u [ ] d 1 du u coshu = 1 u sinhu 1 coshu = 0 u = cothu u u = u 0 u 0 B B 1 u 0 coshu 0 = sinhu B < sinhu 0 ub = coshu B sinhu 0 B u B = 1 u coshu u 1, u u u x = x 1 = c, y 1 = y = A b = 0 aa = coshac ub = coshu ay = coshax S S c dy S = π y 1 + dx = π c dx a 83 u u coshz 1 + sinh z dz